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On the interplay between effective notions of randomness and genericity
In this paper, we study the power and limitations of computing effectively
generic sequences using effectively random oracles. Previously, it was known
that every 2-random sequence computes a 1-generic sequence (as shown by Kautz)
and every 2-random sequence forms a minimal pair in the Turing degrees with
every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We
strengthen these results by showing that every Demuth random sequence computes
a 1-generic sequence (which answers an open question posed by Barmpalias, Day,
and Lewis) and that every Demuth random sequence forms a minimal pair with
every pb-generic sequence (where pb-genericity is an effective notion of
genericity that is strictly between 1-genericity and 2-genericity). Moreover,
we prove that for every comeager , there is some
weakly 2-random sequence that computes some , a result
that allows us to provide a fairly complete classification as to how various
notions of effective randomness interact in the Turing degrees with various
notions of effective genericity